K out of K Extended Visual Cryptography Scheme Based on “XOR”

K out of K Extended Visual Cryptography Scheme Based on

“XOR”

Wanli Dang1*, Mingxing He1, Daoshun Wang2, Xiao Li1

1Department of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China. 2Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China.

*Corresponding author. Tel.: 18781961187; email: [email protected]

Manuscript submitted October 23, 2014, accepted April 10, 2015.

doi: 10.17706/ijcce.2015.4.6.439-453

Abstract: Visual cryptography schemes (VCS) is an encryption technique that utilizes human visual system

in recovering secret image and it does not require any complex calculation. Many VCSs without pixel

expansion have been proposed. But they recover secret image with low contrast due to the stacking

operation. To improve the quality of recovered image, XOR-based VCS has been proposed. However, shares

constructed from XOR-based VCS are random-looking. It suffers a management problem which dealers

cannot visually identify each share. In this paper, we propose an (k, k) extended visual cryptography scheme

(EVCS) by “XOR” operation to solve above mentioned problem. This is implemented by diving the share

image into two parts using a parameter and operating the secret pixels and cover pixels respectively.

Comparing with previous schemes, the qualities of share images and recovered image are improved, and our

scheme also suit the “OR” operation.

Key words: Contrast, pixel expansion, probability, visual secret share, XOR.

1. Introduction

A (k, n) Visual cryptography scheme (VCS) firstly proposed by Naor and Shamir [1] encodes a secret

image into n random-looking shares that appears distinctly different from an “innocent-looking” meaningful

image. The secret can be visually recovered by stacking k or more shares. But such random-looking images

would attract the attention of attackers. One solution to this problem is encoding the secret in meaningful

share images. An important advantage of using meaningful share images is that one can clearly identify each

share by vision [2]. Ateniese et al. [3] have proposed an extended visual cryptographic scheme (EVCS) to

encode a secret image into n meaningful images. This scheme was implemented by concatenating an

extended matrix to each basis matrix. Zhou et al. [4] improved upon Ateniese’s method for dealing with

halftone images designed to make the recovered image less unclear. Chang et al. [5] found a way to hide a

color secret image in two color cover images, but pixel expansion made the share images nine times large

than the original image. Wang et al. [6] presented a general construction method for the extended matrix

and proved the minimum bound of the size of the extended matrix. Their method is simple and easy to

operate. These schemes are realized based on a larger pixel expansion.

Because the visual quality of a recovered image is degraded by a large pixel expansion, most studies try to

reduce the pixel expansion [7]-[10]. For example, Yang et al. [7] proposed a scheme to construct VCS with no

pixel expansion using probabilistic method (PVCS). But these schemes generate shares are random-looking.

International Journal of Computer and Communication Engineering

439 Volume 4, Number 6, November 2015

Chen et al. [11] extended Random Grids-VCS to a user-friendly (2, 2)-VCS, which resolves the pixel expansion

problem and the share meaningless problems by complementary cover images. Ref. [12], [13] continues

Chen’s work proposed (k, k)-EVCS schemes which resolve the pixel expansion and the share meaningless

problems. However, the visual qualities of share images and recovered image of their schemes [11]-[13]

should be further improved.

In order to improve the contrast of recovered image, another model of VCS was proposed by Tuyls [14].

This model constructs an efficient (k, k)-VCS based on “XOR”. A number of XOR-based VCSs were proposed in

[15], [16]. The share images of these schemes are still meaningless. Liu et al. [17] proposed an EVCS for the

even k which based on the “XOR” operation, this can make the share images show the meaningful content,

but there exists extra pixel expansion. So far the study of VCS has been mainly on the OR and XOR operations

[18]-[20]. By a comparison of these two operation, we can find that XOR operation –based VCS have better

parameters in the sense of large contrast and smaller pixel expansion, but the used hardware devices are a

bit more complex than the OR operation-based VCS. Hence, a XOR operation-based VCS which also works

under the OR operation should be a better choice.

To solve the above problems, in this paper, we will propose a (k, k)-EVCS with no pixel expansion by the

“XOR” operation which use the probability method. This scheme improves the qualities of recovered image

and share images and also works under the OR operation. Further, we will give a (k, n) XOR-based

meaningful VCS.

This paper is organized as follows: In Section 2, we introduce some related works and motivation. In

Section 3, we show our scheme and give some theorems about the proposed scheme. In Section 4, we give

some experimental results and compare our scheme with the previous schemes. In Section 5, we conclude

this paper.

2. Related Works and Motivation

In order to better illustrate our scheme, we introduce some notations as Table 1.

Table 1. Introduction of Notations

Notation Used

0B and

1B basis matrices of VCS

H (.) the Hamming weight m pixel expansion for VCS

A the size of the matrix A

0 ,O RP

1 ,( )

O RP the probability of white pixels in the white and black areas when operated by “OR”

operation

0 , X O RP

1 ,( )

X O RP the probability of white pixels in the white and black areas when operated by “XOR”

operation

0 , , 0 , ,c( )

XOR s XORP P the probability of white pixels in the white and black areas of s

fS when operated by

“XOR” operation

1, , 1, ,c ( )

XOR s XORP P the probability of white pixels in the white and black areas of c

fS when operated by

“XOR” operation

0 , , 0 , , c( )

O R s O RP P the probability of white pixels in the white and black areas of s

fS when operated by “OR”

operation

1 , , 1 , , c ( )

O R s O RP P the probability of white pixels in the white and black areas of c

fS when operated by “OR”

operation

2.1. VCS-Based “XOR”

Tuyls et al. [14] gave the contrast and security conditions for XOR-based VCS. And they showed some



construction methods for XOR-based VCS. Then Yang and Wang [10] give a formal definition of conditions for

XOR-based VCS based on the OR-based VCS [1].

Definition 1[10]: The (k, n)-VCS proposed by Shamir which based on “OR” operation is still valid under

the “XOR” operation. And let 'l and 'h represent the whiteness of black and white area in the recovered

image when we stack the shares by “XOR”. Let the notation ( )|i

XOR B r , i=0, 1, denote the XOR-ed vector of

any r rows in Bi .The VCS scheme based on “XOR” operation is considered valid if the following two

conditions are met.

1) [Security]:1 0

( ( | )) ( ( | ))H XOR B r H XOR B r= for ( 1)r k≤ − .

2) [Contrast]:1

( ( | )) ( ')H XOR B r m l≥ − and0( ( | )) ( ')H XOR B r m h≤ − where 0 ' 'l h m≤ ≤ ≤ , i.e.

1 0( ( | )) ( ( | )) ( ' ')H XOR B r H XOR B r h l− ≥ − for r k= .

The first condition (1) assures the perfect secrecy of the scheme. The second condition (2) shows that the

secret image can be visually revealed through their different contrasts of black and white colors. The contrast

1 0[ ( ( ) ( ( )] / ( ' ') /XOR H XOR B r H XOR B r m h l mα = − = − is then defined as the difference in weight between a

white pixel and a black pixel in the reconstructed image [10].

Example 2.1: Consider a (3, 3)-VCS with basic matrices: 0

0 1 1 0

0 1 0 1

0 0 1 1

B

=

, 1

1001

0101

0011

B

=

.

Since0( ( | 2)) 2H XOR B = ,

1( ( | 2)) 2H XOR B = , this satisfies condition (1). Also,

0( ( | 3)) 0H XOR B = and1( ( | 3)) 4H XOR B = , this satisfies condition (2). The contrast is:

1 0[ ( ( | 3)) ( ( | 3)] / 1

XORH XOR B H XOR B mα = − = .

We know, in ref. [1], the VCS can be implemented based on “OR” operation, let ,l h represent the

whiteness of black and white area in the recovered image when we stack the shares by “OR” operation, and

the contrast of this scheme is represented by ( ) /OR h l mα = − . So the following theorem can be obtained.

Theorem 1[10]: The VCS proposed by Shamir [1] which was based on “OR” operation is still a valid under

the “XOR” operation. The difference of white whiteness and black whiteness is ( 1)' ' 2 ( )kh l h l−− = × − . And

( 1)2 k

XOR ORα α−= ⋅ .

Example 2.2 (continuation of Example 2.1): In a (3, 3)-VCS which have been given in example 2.1 with

1, 0, 4h l m= = = and ' 4, ' 0h l= = by “OR” and “XOR” operation, we can obtain ( ) / 1/ 4OR h l mα = − = , then

( 1) (3 1)2 2 (1/ 4) 1k

XOR ORα α− −= ⋅ = ⋅ = .

2.2. PVCS-Based “OR”

The PVCS can solve the problem of pixel expansion for VCS. In PVCS, the probability of white pixels in the

white area is higher than it in the black area of recovered image. Naor and Shamir’s VCS [1] was

implemented by using two basis matrices1B and

0B . A secret pixel is expanded tom sub-pixels and the

number of black pixels for a white and black secret pixel is l and h , they also give the security and

contrast conditions. Yang et al. [7] proposed (PVCS) for binary images with no pixel expansion. The basis

matrices can use all the columns of 1B and

0B . Then we will give the following definition.

Definition 2[7] ：Based on the basic matrix, we can obtain that the probabilities of white sub-pixels in the

white and black areas of recovered image are 0, 1,/ , /OR ORp h m p l m= = . The average contrast is defined as

0, 1, ( ) /OR OR ORp p h l mα = − = − .



In real situation, when two images have the same contrasts, but visual qualities of them are different,

because of the black areas in the recovered image are not reconstructed as 100% black. So they need to

modify the contrast to consist with the real situation.

Definition 3[7]: A new average contrast is defined by their observation of the actual situation,

0, 1, 1,( ) / (1 )OR OR OR ORp p pα = − + .

2.3. Motivation

In order to improve the contrast of recovered image, “XOR” operation of VCS was proposed. The VCS

proposed by Shamir which based on “OR” operation is still valid under the “XOR” operation. When we

operate secret image by probabilistic method based on “XOR” in ( , )k k VCS− , we will find that every pixel

of secret image can be correctly reconstructed.

Example 2.3: Construct basis matrices for (2, 2)-VCS, the basis matrices B0 and B1 are shown below.

0

0 1,

0 1B

=

, 1

1 0,

0 1B

=

.

We operate secret image by probabilistic method based on “XOR”, the experimental result is shown below.

Our all experiments use secret image is following, where “ ⊗ ” represents the “XOR” operation (See Fig. 1 and

Fig. 2).

Fig. 1. The experiments for “⊗”operation. Fig. 2. The experiment for our method.

This can correctly reconstruct secret image and with no pixel expansion, however share images are

looking-random, these suffer from a management problem-dealers cannot identify shares. Conventional

EVCSs designed basis matrices to encode each secret pixel as a block of sub-pixels, which leads to a pixel

expansion problem. Chen and Tsao [11] extended Random Grids (RG)-based VCS to a user-friendly (2, 2)-VCS,

which resolves the pixel expansion problem and the share images are meaningful by complementary cover

images. The contrast of the recovered image and meaningful shares is a tradeoff made by adjusting

parameters during construction of the method. If we use their method to divide share images into two parts,

a part is used to operate secret pixel, another is used to embed cover pixels, when we stack the shares by

“XOR”, we can implement the result as follows (The operation method will be presented in Section 3).

The visual quality of shares and recovered images can be improved compared with Chen and Tsao’s

method. More details discuss in Section 4. We know XOR-based VCS can improved the qualities of images, but

have a bit more complex than the OR operation-based VCS. So an EVCS that can work both under the OR and

XOR operation will be a better choice. In this work, we guess whether our method can be extended to a



general (k, k)-scheme for sharing binary secret images by “XOR” operation. And the proposed scheme also

suit the “OR” operation. The objectives of the study include solving the pixel expansion, providing an

adjustable contrast for recovered image and meaningful shares to satisfy dealers’ requirements, and

improving the qualities of the recovered images and share images.

3. The Proposed Scheme

In this section, we propose a (k, k)-EVCS scheme for binary images by the “XOR” operation. In Section 3.1,

we describe the construction procedure for the shares. Next we analyze the proposed scheme with the help

of the previous results which are presented in Section 2.

3.1. The Process of Encryption

The proposed extended visual cryptographic scheme based on “XOR” extends from user friendly RG-based

VCS proposed by Chen and Tsao [11]. The proposed (k, k)-EVCS based on “XOR” operation can implement the

meaningful shares and generate the share images and recovered image with no pixel expansion. In the

encoding phase, a secret image S and a cover image C, both with the size of p q× are encoded in k

meaningful shares Sf (f=1,…,k) with the same size of S.

The process of encoding a pixel is demonstrated in Fig. 3. One should select a parameter β , where β is

used to make a trade-off visual qualities between share images and recovered image and choose an

operation between a secret pixel S (i, j) and a cover pixel C (i, j) (1 ,1 )i p j q≤ ≤ ≤ ≤ . While the results of

stacking all share images 1 2 kS S S⊗ ⊗ ⊗⋯ will reveal the content of the secret image S.

Fig. 3. The process of encryption.

In Fig. 3, the equation of (.)pF will be designed as follows: when ( , ) 0,S i j = we will randomly select

column from B0; when ( , ) 1,S i j = we will randomly select column from B1. This is the probability method,

the basic matrices are B0 and B1. And the Equation of (.)CF will be designed as follows:

( , ) 1, if ( , ) 1( )

( , ) 0, if ( , ) 0

f

C

f

S i j C i jF

S i j C i j

= =⋅ = = = (a)

In the design of the equation (.)CF , ( , )fS i j denotes the each share image pixel (1 ,1 )i p j q≤ ≤ ≤ ≤ .

This can implement the meaningful share images. In our scheme, the designs of the encoding process and

the cover image embedding process for (k, k)-EVCS are separated. With the parameter β , the proposed



scheme provides an adjustable contrast for meaningful shares and recovered image to satisfy dealers’

requirements.

3.2. Construction of Shares

For easily illustrating the proposed scheme, a definition and notations are given first.

In the following Algorithm 1, the function ( )X F M← randomly selects an unrepeatable pixel position

( , )i j from the input image M and stores ( , )i j into the sequence X, where 1 ,1i p j q≤ ≤ ≤ ≤ . fS

represents share image which shows the content of cover image. c

fS represents the area of fS when we

operate the pixels of cover image and store them in the corresponding positions of the share image fS .

Similarly, s

fS represents the area of fS when we operate the pixels of secret image and store them in the

corresponding positions of the share image fS . Thus , ,c s

f f fS S S satisfy: c s

f f fS S S= ∪ and c s

f fS S∩ = ∅ .

When we stack the shares, 1 2 kS S S⊗ ⊗ ⊗⋯ , the secret information will be revealed. Next we will give the

construction of (k, k)-EVCS by “XOR” operation.

Let B0 and B1 be the basis matrices for a binary VCS with pixel expansion m and contrast α , Boolean

XOR operation. Our proposed (k, k)-EVCS takes two input images: one is the secret image S , the other is

cover image C .

Next, we will give an example which can help us to understand the Algorithm 1.

Algorithm 1:

Input: A secret image { ( , ) |1 ,1 }S S i j i p j q= ≤ ≤ ≤ ≤ ,

a cover image { ( , ) |1 ,1 }C C i j i p j q= ≤ ≤ ≤ ≤ . And a

parameter β , 0 1β≤ ≤ .

Output: k shares with meaningful ( 1, , )fS f k= ⋯

{ ( , ) | 1 ,1 }f fS S i j i p j q= ≤ ≤ ≤ ≤ .

Step1: Generate a random bit α which

satisfies: ( 0) , ( 1) 1p pα β α β= = = = −

Step2: If 0α = , we operate the pixels of secret

image ( , )S i j , and store it in ( , )s

fS i j of the

shares. Namely, ( , ) ( ( ( , )))s

f pS i j F F S i j← .

Step3: If 1α = , we operate the pixels of cover

image ( , )C i j , and store it in ( , )c

fS i j of the

shares. Namely, ( , ) ( ( ( , )))c

f CS i j F F C i j← .

Step4: If k is even, Output fS ( 1, , )f k= ⋯

Step5: If k is odd, when the generated k pixels

( ( , ) 0fS i j = , 1, ,f k= ⋯ ) for the areas of

c

fS are all white, we randomly choose

a ( , )c

fS i j and set it to 1.

Step6: Output (1 2, , ,

kS S S⋯ )

Example 3.1: A (3, 3)-EVCS scheme, the main codebook B0 and B1 are 0

0110

0101

0011

B

=

and 1

1001

0101

0011

B

=

and



0.5β = . Fig. 2 illustrates the encryption process in the scheme.

(a) (b) (c) (d)

(e) (f) (g) (h)

Fig. 4. An example of (3, 3)-EVCS with β=0.5: (a) Secret image; (b) Cover image; (c) Coordinates; (d)

Encryption decision, Bi(i=0, 1): the selected encoding for secret image, C: for cover image; (e) Share 1; (f)

Share 2; (g) Share 3; (h) Share1⊗Share2⊗Share3.

Fig. 4 presents the process of encoding secret image and embedding cover image. And “1” represents the

black pixel, “0” represents the white pixel. The black blocks represent the area of c

fS , the white blocks

represent the area of s

fS . Fig. 4(e)-Fig. 4(g) illustrate share1-share3. We randomly select column from B0 to

encrypt pixels for secret image at coordinates 2, 5, 10, 14. Similarly, we randomly select column from B1 to

encrypt pixels for secret image at coordinates 4, 7, 12, 15. Then we embed pixels for cover image at

coordinates 1, 3, 6, 8, 9, 11, 13, 16 based on the step 3 of algorithm 1. Observation of the (h), the black blocks

are black, so the cover image does not leave any trace in the recovered image. And obviously, the white pixels

in the white area more than in the black area of recovered image, so the secret image can be revealed.

3.3. Analysis of Images’ Contrasts

In this section, we will analyze the contrasts of recovered image and share images with the help of the

previous results which are presented in Section 2.

3.3.1. Contrast of recovered image

We use the probabilistic VCS method for S(i, j). Before proving the theorem of the contrast of recovered

image, we need to present three lemmas.

Lemma 1 [1]: For a probabilistic (k, k)-VCS, when we stack the shares by “OR” operation. We can obtain

1

0, 1 / 2kORP −= ,1, 0ORP = and 11/ 2kORα −= .

Lemma 2: For a probabilistic (k, k)-VCS scheme, when we stack the shares underlying “XOR” or “OR”

operation. We can obtain ( 1)

2k

XOR ORα α−= ⋅ .

Proof: Because of the probabilistic (k, k)-VCS’s implementation can use all the column of0B and

1B . So the

result of the theorem 1 also suit for the Lemma 2.

Lemma 3: For a probabilistic (k, k)-VCS scheme, when we stack the shares underlying “XOR” or “OR”

operation. We can obtain: ( 1)

0, 0,2 k

XOR ORp p−= ⋅ ,1, 1, 0XOR ORp p= = .

Proof: Form ref. [6], we know the property of the basis matrices of the (k, k)-VCS: the number of “1” is

even for every column of 0B and the number of “1” is odd for every column of

1B and 1

0 1 2 kB B −= = .

By the property of XOR operation (0 for even “1” and 1 for odd “1”), when any k rows of 0 1( )B B under the

“XOR” operation, we can obtain: 0, 1XORP = and

1, 0XORP = . From Lemma 2, we have: 1

0, 1/ 2kORP −= and



1, 0ORP = . Thus ( 1)

0, 0,2 k

XOR ORp p−= ⋅ and 1, 1, 0XOR ORp p= = .

Now we state the Theorem 2 as follows.

Theorem 2: The average contrast of the stacking result 1 2 kS S S⊗ ⊗ ⊗⋯ of Algorithm 1 is:

( ) (0 1)F mα β β= ≤ ≤ .

Proof: From the encoding process of Algorithm 1, we know that ( , )fS i j is from ( , )S i j with

probability β and from ( , )C i j with probability (1 )β− . Thus we have the following two cases:

1) ( , )fS i j is from ( , )S i j with probability β . We consider the area of s

fS . When we stack share

images underlying “OR” operation, from Lemma 1, we can obtain:

1

0, , / 2kOR sp β −= , 1, , 0OR sp = . (1)

From lemma 4, we can obtain:

0, , 1, ,, 0XOR s XOR sp pβ= = (2)

2) ( , )fS i j is from ( , )C i j with probability 1 β− . We consider the area of c

fS . By the property of

XOR operation (0 for even “1” and 1 for odd “1”) and the step 3, 4 of Algorithm1, we can obtain:

When k is even:

0 , , 1, ,1 (1 ), 1 (1 )XOR c XOR cP Pβ β= ⋅ − = ⋅ − (3)

When k is odd:

0 , , 1, ,0 (1 ), 0 (1 )XOR c XOR cP Pβ β= ⋅ − = ⋅ − (4)

Thus the average contrast of recovered image is:

0 , 1,( )

F XOR XORm P Pα = − 0, , 0 , ,c 1, , 1, ,c( ) ( )XOR s XOR XOR s XORP P P P= + − + β=

3.3.2. Contrast of share images

Property 1: Given a (k, k)-VCS, let its basic matrices be ( 0,1)iB i = , we have:

1) 0 1( | ) ( | )H B i H B i= , i represents every row of

iB .

2) 0 0 1 1( | ) ( | ), ( | ) ( | )H B i H B j H B i H B j= = , i j≠ .

3) The number of “1” is equal to the number of “0” for every row of ( 0,1)iB i = .

Theorem 3: For any f satisfy 1 f k≤ ≤ , the average contrast of share images fS is: (1) when k is

even, ( ) 2(1 ) / (2 )S mα β β= − + ; when k is odd, ( ) [2 ( 1) (1 )] / [ (2 )]S m k kα β β= ⋅ − ⋅ − ⋅ + .

Proof: From the Algorithm 1, we know that the proposed scheme procedure is designed differently with

the different k (odd or even).

1) whenk is even



� ( , )fS i j is from ( , )S i j with probability β . We consider the area of s

fS . From the property 1, the

probabilities of white pixel in the white and black areas of s

fS are:

0, 1,(1 / 2) , (1 / 2)s sp pβ β= ⋅ = ⋅ (5)

where ( 0,1)i,sp i = represents the probability of white pixels in the white and black areas of s

fS .

� ( , )fS i j is from ( , )C i j with probability 1 β− . We consider the area of c

fS . From the step 3 of

Algorithm1, the probabilities of white pixels in the white and black areas of c

fS are:

0 , 1,1 , 0c cp pβ= − = (6)

where ( 0,1)i,cp i = represents the probability of white pixels in the white and black areas of c

fS , thus we

can obtain the probability of white pixels in the white area and black area of fS are:

0 0, 0, (1/ 2) (1 )s cp p p β β= + = ⋅ + − (7)

1 1, 1, (1 / 2) 0 (1 / 2)s cp p p β β= + = ⋅ + = ⋅ (8)

From Equations (7), (8) and definition 3, we can obtain the average contrasts of share images are:

0 1 1( ) ( ) / (1 ) 2(1 ) / (2 )S m p p pα β β= − + = − +

2) when k is odd

� ( , )fS i j is from ( , )S i j with probability β . We can obtain that 0, 1,(1 / 2) , (1 / 2)s sp pβ β= ⋅ = ⋅ .

� ( , )fS i j is from ( , )C i j with probability 1 β− .

The share pixels are generated by Step 4 of Algorithm 1. When ( , ) 0C i j = , the probability of

generating white pixel of ( , )fS i j is ( 1) /k k− . We can obtain, the probability of white pixels in the white

(resp. black) area of CX is:

0, [( 1) / ] (1 )cp k k β= − ⋅ − , 1, 0cp = . (9)

Thus we can obtain that the probabilities of white pixels in the white and black area of fS are:

0 0 , 0 , (1 / 2) [( 1) / ] (1 )s cp p p k kβ β= + = ⋅ + − ⋅ − ;

1 1, 1, (1 / 2) 0 (1 / 2)s cp p p β β= + = ⋅ + = ⋅ (10)

From Equation (10) and definition 3, we can obtain that the average contrasts of share images are:

0 1 1( ) ( ) / (1 ) [2 ( 1) (1 )] / [ (2 )]S m p p p k kα β β= − + = ⋅ − ⋅ − ⋅ +



3.4. Analysis of Algorithm1’s Security

Theorem 4(security): For the proposed scheme, when we stack the share images,1 2 kS S S⊗ ⊗ ⊗⋯ , we

can obtain1 0

( ( | )) ( ( | ))H XOR B r H XOR B r= , ( )r k< . In other words, the probabilities of white sub-pixels

in the white and black areas of recovered image are equal.

Proof: From the encoding process of Algorithm 1, the designs of the encoding secret image and

embedding cover images are separated.

1) When ( , )fS i j is from ( , )S i j ，from definition 1, we know the fact that the Hamming weight of the XOR

of any t, t<k, rows are the same in the basic matrices ( 0,1)iB i = . So the probability of white sub pixels

in the white and black areas of s

fS is equal.

2) When ( , )fS i j is from ( , )C i j , since the ( , )fS i j dose not depend on the secret pixel which is from

( , )C i j . So the shares of c

fS will not leak the secret information.

Therefore, the security condition is ensured.

3.5. Recognition of Small Areas in Secret Image

In a deterministic (k, k)-VCS scheme, there exists a difference between a recovered black pixel and a

recovered white pixel. But in probabilistic VCS scheme, the difference is not maintained. In ref. [14], we know

that the RG-based VCS is a special case of the probability VCS. So Chen’s scheme also exists the recognition of

small areas in the recovered image. In this section, we will discuss the recognition of small areas for our

proposed scheme.

In ref. [7], let N be the total number of pixels in an area of secret image, NS be the number of white pixels

in that area of the recovered image. The probability of each pixel being white is 0 ,ORp (resp. 1,ORp ) in a

white (Black) area of secret image. In ref. [6], the NS obey the normal distribution. So it gets the lower

bound of N that one can distinguish the black for the Prob-VCS scheme as the following:

2

0, 0, 1, 1, 0, 1,9[( (1 ) (1 )) / ( )]OR OR OR OR OR ORN p p p p p p d> − + − − − 0, 1,(0 ( ))OR ORd p p≤ ≤ − (10)

For the proposed scheme, our recognition of small areas in the secret image is similar to Yang. So we can

obtain:

2

0, 0, 1, 1, 0, 19[( (1 ) (1 )) / ( )]XOR XOR XOR XOR XOR ,XORN p p p p p p d> − + − − − 0 , 1,(0 ( ))XOR XORd p p≤ ≤ − (11)

From Lemma 3 and Algorithm 1, we can obtain:

1

0, 0,2k

XOR ORp pβ−= ⋅ ⋅ ,

1

1, 1,2k

XOR ORp pβ−= ⋅ ⋅ (12)

In a (k, k)-VCS scheme, 1, 0ORp = ,1

0 , 1 / 2 kORp −= ,

From Equation (11), (12), the N satisfies the following equation:

29[ (1 ) / ( )] (0 )N d dβ β β β> − − ≤ ≤

That is, the β determine the lower bound of N, the recognition of small areas in secret image do not rely

on the basis matrices, the reason that is a white secret pixel must be decoded a white pixel and a black secret



pixel must be decoded a black pixel when we operate the shares by “XOR” in (k, k)-VCS. Many times, the share

information is not more important than the secret information, so that we have 0 .5 1β≤ < .

4. Experiment and Comparison

This section, we first present the experimental results for our scheme and the previous schemes. The

comparison experimental result is summarized as Table 2. Second, we compare the proposed scheme with

some related schemes in two aspects: (1) The contrasts of share images and recovered image; (2) Comparing

the proposed scheme with the previous in some typical aspects. Comparison results respectively by Tables 3

and 4.

Table 2. The Experimental Results for Some Schemes

Schemes

with 0.5β = Share 1 Share 2 Recovered

image

Chen and

Tsao [7]

3 / 11s

α = 3 / 11s

α = 1 / 9F

α =

Guo [8]

1 / 5

sα = 1 / 5

sα = 1 / 4

Fα =

Our

scheme

2 / 5S

α = 2 / 5S

α = 1 / 2F

α =

In the Table 2, we choose the parameter 0 .5β = . We can see, In Chen and Tsao [11]’s scheme, the visual

qualities of their share images and recovered image are so far worse than the ones obtained with our

scheme and Guo [12]’s scheme. The main reason is that in their scheme, the black areas in the recovered are

not reconstructed as all black. And obviously, our scheme outperforms Guo’s scheme in terms of the qualities

of share images and recovered image.

From Table 3, we observed that as β increases for our scheme, the contrast of recovered image increase

while the contrasts of share images decrease (the experimental results present in Appendix A). By observing

Table 2 from left to right, we observed that when the value of β is certain, the proposed scheme

outperforms the schemes proposed by Chen [11] and Guo [12] in terms of the contrasts of share images and

recovered image.

Table 4 presents the results of comparison between our scheme and the previous schemes. From Table 4,

we can see that comparing with the previous scheme, the quality of the share images and the recovered

image are improved. And the proposed scheme has no pixel expansion with meaningful share images. But

our scheme have a bit more complex than the OR operation-based Schemes. So sometimes, a scheme that

can work both under the OR and XOR operation will be a better choice. When we consider (k, k)-EVCS with

even k, we can use a pair of complementary cover images, when k is odd, and our input is not change (the

experimental results present in Appendix B). The proposed scheme was based on “XOR” operation is still a

valid under the “OR” operation.



Table 3. The Contrasts of Share Images and Recovered Image That Are Generated by (2, 2)-EVCS with β = 0,

1/4, 1/2, 3/4, 1

β

Ref.[7](model 1) 1ρ = Ref.[8] Our scheme

1

3S

βα −= 3

F

βαβ

=−

1

2S

βαβ

−=+

1

2Fα β= 2 (1 )

2S

βαβ

−=+ F

α β=

0 1/3 0 1/2 0 1 0

1/4 1/4 1/11 3/9 1/8 2/3 1/4

1/2 1/6 1/5 1/5 1/4 2/5 1/2

3/4 1/12 1/3 1/11 3/8 2/9 3/4

1 0 1/2 0 1/2 0 1

Table 4. Comparison with Previous Results

Schemes Pixel

Expansion

Meaningful

Share Decryption

Complexity

decryption

Visual

Quality

Type of

VSS

Ref.[3],[4] Yes Yes OR O(1) Medium (k, n)

Ref.[11] Yes No XOR O(k) High (k, n)

Ref.[6] No No OR O(1) Medium (k, n)

Ref.[5] Yes Yes XOR O(k) High (k, n)

Ref.[7] No Yes OR O(1) Low (k, k)

Ref.[8] No Yes OR O(1) Medium (k, k)

Our No Yes XOR O(k) High (k, k)

For example: The proposed (2, 2)-EVCS with 0.5β = , our experimental results are shown in Fig. 5:

(a) (b)

(c) (d)

Fig. 5. The experiments with (2, 2)-EVCS: (a) share1, (b) share2, (c) share1⊗share2, (d) share1⊗share2.

where the “ ⊕ ”represent the “OR” operation, we can see, our proposed scheme not only suit the “XOR”

operation, but also suit the “OR” operation.

5. Conclusion

This paper proposed a (k, k)-EVCS scheme based on “XOR” operation. When we use a pair of

complementary cover images for the even k, our scheme is still valid under the “OR” operation. The visual

quality between share images and recovered image can be adjusted to be more friendly for the dealer by

different β . Comparing with the previous scheme, the quality of the shadow images and the recovered

image are improved.

Appendix

Appendix A



This section gives the experimental results for Algorithm 1 of (2, 2)-EVCS with β =0, 0.25, 0.5, 0.75 and 1.

By observing Table 5 from top to bottom, it is convenient to see that as β increases, the quality of recovered

image increase while the qualities of share images decrease, consistent with the above inference from Table

3.

Table 5. The Experimental Results for (2, 2)-EVCS with Different β β Share 1 Share 2 Recovered image

0

1/4

1/2

3/4

1

Appendix B

This section gives the experiments of (3, 3)-EVCS with β = 0.5, we can see that when k is odd, the proposed

scheme was based on “XOR” operation is still a valid under the “OR” operation. (See Fig. 6).

(a) (b) (c) (d) (e) (f) (g)

(h) (i) (j) (k)

Fig. 6. The experiments with (3, 3)-EVCS: (a) share 1, (b) share 2, (c) share 3, (d) share1⊗share2, (e)

share1⊗share3, (f) share2⊗share3, (g) share1⊗share2⊗share3, (h) share1⊗share2, (i) share1⊗share3, (j)

share2⊗share3, (k) share1⊗share2⊗share3.

Acknowledgments

This research was supported in part by the National Key Technology R.D Program of the Ministry of

Science and Technology, (No.2011BAH26B00), International Cooperation Project of Sichuan Province

(2009HH0009), Chunhui project of the Ministry of China (Z2014045), and Sichuan Province Key Discipline



Construction Project (ZD0802-09-1), and the Graduate Innovation foundation ycjj2014040.

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Wanli Dang was born in Shanxi Province, China, in 1990. She received the B.S. degree

from the Department of Mathematics at Xianyang Normal University, China, in 2012. She is

currently a M.S. candidate at the School of Mathematics and Computer Engineering, Xihua

University, China. Her current research interests conclude cryptography and applied

mathematics.

Mingxing He received the M.S. degree from Chongqing University and the Ph.D. degree

from Southwest Jiaotong University, China, in 1990 and 2003, respectively. He is a member

of the IEEE and ACM, a senior member of CAAC. He is currently a professor and the

director of the Key Lab of Cyberspace Security Insurance of Sichuan, Xihua University. He

has published over 80 research papers in refereed journals and conferences. He has been a

reviewer for several international academic journals. His research interests include

information security and cryptography.

Daoshun Wang received his B.S. degree from the Department of Mathematics, Lanzhou

University, China, in 1987, and a Ph.D. degree from the Department of Mathematics,

Sichuan University, China, in 2001. He is currently an associate professor of the

Department of Computer Science and Technology, Tsinghua University. He is a member of

IEEE. His research interests include visual cryptography, digital watermarking and label

anti-counterfeit.

Xiao Li was born in Sichuan province, China, in 1972. He received the M.S. degree from

Xihua University in 2005. He is an associate professor and a graduate students supervisor

of Xihua University. He has published over 10 research papers in refereed journals and

conferences. His current research interests include information security and cryptography.

Documents

K out of K Extended Visual Cryptography Scheme Based on “XOR”